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I think that very often the idea of 'constraint' appears in mathematics. For example, when a triangle is considered, 3 points are constrained not to be co-linear, and then we try to discover the implications of this initial constraint, when the rules of logic are applied to it.

Doing this, we discover implications which, although present from the beginning, are not immediately apparent, so essentially we discover what we have already assumed without knowing it.

So in this very general setting, my 2 questions are:

1) What is the most general way to define and separate "the rules of logic" from "the things to which the rules are applied" ?

2) If the above separation can be made, is there some "minimal constraint" on the things upon which logic is applied, enabling us to see the workings of logic in the most pure form, i.e. with minimal inteference with the properties of things upon which logic acts?

For example, suppose we start with the constraint: let S be an object. Nothing else is assumed. Is there something meaningful and "non-trivial" implied by this constraint?

Maybe some of the words used above are not as well-defined as needed, but i leave them unchanged hoping that the direction towards which i'm asking for guidance is not very unclear.

EDIT: (too long for a comment) based on the answer of Mauro ALLEGRANZA. As humans we have the innate ability to separate words and construct sentences with persistent meanings. This seems to me a very great capability, which pre-exists even relative to other very basic "mathematical" ones, e.g. the capability to duplicate things, to count things, to conceive concepts like "for every", to conceive the set of natural numbers...
So is there a universally accepted foundation of logic which clearly separates the ultimate basic concepts from all the rest?
Am i wrong in thinking that all known formalizations address concepts already well-understood relative to the still obscure foundations of logic, (e.g. the border between physical language and natural numbers) resulting in an ever-present foundational mist?

EDIT 2 (motivated by the answer of Bumble): I wonder whether logical constants include (besides concepts as 'and','or','if' etc.) the natural numbers too. They seem absolutely necessary for "primitive" logical processes like the separation of words and the formulation of sentences. If this is true, then the clear separation of "logic" from "objects upon which logic acts" becomes problematic, as natural numbers appear to be some sort of "common ground". Is it the only common ground? Could geometric shapes also exist in logic? (for example, we have the ability to spot "cyclical reasoning". Is this just a play of words, or is it produced by the idea of a cycle, or a closed curve in general?)

  • Well, "object" isn't defined in math unless it's in the context of category theory, and then it means that it's in some category and has at least an identity morphism. In set theory it doesn't mean anything at all. But if we say it's a set, we're saying it has mathematical existence but possibly no other property. For example the axiom of choice gives us sets whose only attribute is that they exist; we can't list their elements or define their elements by any procedure or formula. Is all this what you're getting at? – user4894 Jun 26 at 8:53
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    "Pure logic" is just the standard predicate calculus, for example, any vocabulary and axioms ("constraints") added to that would be non-logical. There is no "minimal constraint", but a common very modest extension is to add equality symbol = with the standard properties as axioms, mereology studies extensions that also add "part of" symbol. "Let S be an object" is not a constraint since "object" can mean whatever one wishes, unless some axioms about it are also added. – Conifold Jun 26 at 11:28
  • Look at many sorted logic (the sorts are the sets of what you call "objects") – Rusi Jun 26 at 12:17
  • @Conifold: I wonder whether "Let S be an object" (meaning 1 object and not e.g. 2) presupposes our ability to count, and consequently the set of natural numbers. Can the set of natural numbers be separated from the rules of logic? – exp8j Jun 26 at 12:23
  • What "object" presupposes and stands for will have to be spelled out and stated axiomatically, it can not be left open-ended, relying on implicit intuitions, informal "meaning" of objects, etc. Look at formal ontology, e.g. formalization by Smith. Non-logical axioms for natural numbers are spelled out in Peano arithmetic. – Conifold Jun 26 at 17:21
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The "rules of logic" are the object of study of formal logic and mathematical logic.

They define languages and proof systems, like e.g. predicate calculus, that are "applicable" to any topics whatever.

The so-called "laws of logic" are formulas that are true irrespective of any possible interpretation, i.e. they hold in every interpretation.

In this sense, the laws of "pure" logic are applicable to every domain of discourse consisting of objects whatever.

In addition to the study of "pure" predicate logic, we are interested to add to the "logical engine" suitable non-logical constants, like ("in"), the binary relation of set theory, or + and × ("plus" and "times"), the basic arithmetical operations, with suitable axioms that govern their behaviour.

Thus, according to the specific mathematical symbols and axioms introduced, we have different mathematical theories; when the collection of axioms is the first-order version of Peano's axioms, we have PA, i.e. first-order theory of arithmetic.

The same for ZF, i.e. Zermelo-Fraenkel Set Theory.

In this sense, first-order theory of arithmetic and set theory are "applications" of logic to specific domains of inquiry (or universes).

We may say that formal mathematical theories are the study of topics where the universe is "constrained" to some specific properties : those defined by the non-logical symbol used (, +) and by the specific mathematical axioms postulated for the theory.

  • Thank you for answer and for giving me the opportunity to briefly express some further difficulties i've always had in my attempts to study logic, added as an edit to the original question. – exp8j Jun 26 at 11:42
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1) What is the most general way to define and separate "the rules of logic" from "the things to which the rules are applied" ?

The basic distinction is obtained by the definition of propositions and of the logical operations done on them. "Separation" is achieved in the sense that only the truth of the propositions is relevant to the way the operations work.

2) If the above separation can be made, is there some "minimal constraint" on the things upon which logic is applied, enabling us to see the workings of logic in the most pure form, i.e. with minimal inteference with the properties of things upon which logic acts?

The "things upon which logic is applied" being the truth of the propositions, it is necessary to identify all constraints that may exist between the truth of the different propositions to which logical operations are going to be applied.

This normally requires to spell out the propositions themselves, and to define all concepts that are relevant to their truth. So, to apply logic to the proposition "S is an object", in your example, it would be necessary to define first the concept of "object".

This can only work essentially on abstractions because most real-life situations and macroscopic objects cannot be so exhaustively described. Thus, deductive logic isn't generally conclusive outside of its application to abstractions, some areas of physics and ... personal beliefs.

When this is done, there is no "interference" outside that mediated by the truth of the different propositions involved.

So is there a universally accepted foundation of logic which clearly separates the ultimate basic concepts from all the rest?

The abilities that it is necessary to have to perform the required logical analysis will depend on the particular propositions involved. We can choose the propositions involved so as for example have no need for counting (e.g. A and B implies A). Beside the logically operational semantics of the conjunction, "A and B implies A" only requires to be able to distinguish between "A" and "B". No notion of counting or of number is involved. And, presumably, while counting requires the ability to distinguish, distinguishing does not require counting.

Thus, it is not the logical operations that presuppose counting.

The analysis of propositions is distinct from the logical calculus performed on their truth values. However, the analysis comes before the application of the operations. Any ability necessary to perform the analysis is therefore presupposed. However, it is presupposed by the analysis of the propositions involved, not by the application of logical operations.

Thus, the analysis of propositions involving sets, for example, presupposes an ability to analyse the relations between the kinds of set involved.

And the analysis of propositions involving numbers requires an ability to analyse the relations between the kinds of number involved.

No notion of number is required for "A and B implies A". A conjunction requires the ability to distinguish between a number of conjuncts. It does not involve and therefore require any counting or any notion of number.

Thus, the implication "A1 and A2 and A3 ... and A1000 implies A567" does not require any counting or notion of number. All is required is the semantics of "and" and the ability to distinguish A597 from the other premises.

Thus, where counting is required, it is required by the semantics of the propositions involved.

  • By "things upon which logic is applied" i mean not only the truth-values of propositions of logical calculus, but also the "things" used to build these propositions, like numbers & geometric shapes. But i am puzzled by the fact that e.g. numbers seem to get involved not only as "objects of study" but also as logical operations! E.g. in order to say that "A and B implies A", i think the concept "natural number" must already exist in our reasoning. Otherwise how could we separate these 5 words, match the 2 occurrences of A and combine all into 1 persistent meaning? – exp8j Jun 27 at 11:50
  • A comment on the ability to distinguish between a number of conjuncts, mentioned in your concluding remarks: I think that "ability to distinguish A from B" is the same as "ability to perceive the number 2, abstractly". Do you find some error in this? And truly I would be glad to dismiss my original questions as nonsense, but i'm still not convinced:) – – exp8j Jun 27 at 18:27
  • @J.Avaris I don't think 'that "ability to distinguish A from B" is the same as "ability to perceive the number 2, abstractly"'. You might as well say that ability to distinguish A from B from ... from N is the same as the ability to perceive a number N abstractly, whatever the value of N. Distinguishing isn't counting. – Beanluc Jun 27 at 18:27
  • @Beanluc: I did not generalize to N > 2. But for N=2, i think that pair-wise distinction is the same as perception of the number 2. For N > 2, i agree that distinguishing isn't counting, if distinguishing is done pair-wise. It's unclear to me how to dinstinguish N things simultaneously. – exp8j Jun 27 at 18:55
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Here are the questions:

  1. What is the most general way to define and separate "the rules of logic" from "the things to which the rules are applied" ?

The following definition of "theory" from Wikipedia may help clarify the separation:

A theory about a topic is usually a first-order logic together with a specified domain of discourse over which the quantified variables range.... [my emphasis]

The first-order logic would correspond to "the rules of logic"; the domain of discourse to "the things to which the rules are applied".

This Wikipedia article also provides a set of definitions of the rules or concepts used in first-order logic. This could be viewed as a "general way" to do this. Textbooks on deductive logic will also define these (perhaps slightly differently).

Two textbooks from the Open Logic Project provide more depth. The forallx text provides an introduction and The Open Logic Text provides a more advanced presentation.

  1. If the above separation can be made, is there some "minimal constraint" on the things upon which logic is applied, enabling us to see the workings of logic in the most pure form, i.e. with minimal interference with the properties of things upon which logic acts?

Assuming that the domain corresponds to the things upon which this deductive logic is applied, the minimal constraint is that these things have to be individuated enough so that they can be members of a domain or set.


The above applies to deductive logic. There is also non-deductive logic. Wikipedia describes defeasible reasoning as

In logic, defeasible reasoning is a kind of reasoning that is rationally compelling, though not deductively valid.

This is also called "presumptive" reasoning and this is sometimes studied through the concept of "argumentation schemes".


Wikipedia contributors. (2019, January 28). Defeasible reasoning. In Wikipedia, The Free Encyclopedia. Retrieved 12:41, June 26, 2019, from https://en.wikipedia.org/w/index.php?title=Defeasible_reasoning&oldid=880695250

Wikipedia contributors. (2019, June 16). First-order logic. In Wikipedia, The Free Encyclopedia. Retrieved 12:41, June 26, 2019, from https://en.wikipedia.org/w/index.php?title=First-order_logic&oldid=902047532

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If I understand your question correctly, you are asking in effect how do we distinguish logic from non-logic? Logical expressions give rise to valid arguments and logical truths, that is, arguments where if the premises are true it is impossible for the conclusion to be false, and truths such that there is no way for them to come out as false. But this prompts us to enquire what it is about such arguments or sentences that warrants us to say that they determine what is possible, rather than merely what is true.

Consider an example. A simple valid argument is: grass is green and snow is white; therefore grass is green. This argument remains valid whatever we substitute for 'grass is green' or 'snow is white'. We can even emphasise the point by using propositional symbols: P and Q; therefore P. But we cannot substitute the 'and' with something else. It would not do to write: P or Q; therefore P. Our understanding of the validity of the argument depends on holding some features of the sentences constant, while allowing others to be variable. Those features of sentences that need to be held constant to support our understanding of what is valid and what is not are commonly called the logical constants. They include 'and', 'or', 'not', 'if', 'every', 'some', and can be extended to include 'equals', 'is a member of', 'is part of', 'contains', etc.

If we ask what justification there is for considering some terms to be logical constants and not others, then several different answers have been given. One is to say that logic is by its nature formal: it is concerned with separating the form of a proposition from its content. The logical constants are then those terms that serve this formal role. Another approach is rooted in grammar. Some words allow us to build long complex sentences from simple ones in such a way that we can systematically understand the complex sentences in terms of the simple ones. The logical constants are then the connectives that support this function. Another approach is to say that what distinguishes logic from non-logic is that logic is 'topic-neutral', i.e. it is not about any particular subject matter but applies universally. The logical constants are then those terms that allow sentences to remain true under all uniform substitutions of the words that have a subject matter. Another common approach is to introduce the technical concept of 'interpretation', which is a function that assigns referents to names, classes to predicates and truth values to propositions. On this approach, a valid argument is one such that there is no interpretatation under which the premises are true and the conclusion false. The logical constants are those terms that are not a feature of the interpretation function. Another approach emphasises that logical constants are not sensitive to the identity of particular objects. Logical truths are then those that remain true under all permutations of the objects within our domain, and the logical constants are the terms that allow us to express such truths.

There remains an issue that any of the above approaches could be considered question-begging. We still rely upon a fundamental understanding of what things or properties should be treated as variable and what as constant. This makes our collection of logical constants look like a laundry list rather than a principled distinction. Some have pursued the idea of identifying the logical constants by the inferential relationships they warrant. Others have questioned whether logical validity is essentially formal.

  • I'd be interested in some references for the third paragraph: ideas for distinguishing logical in a principled way. Big names were skeptical, Tarski:"I also consider it quite possible that we shall be compelled to regard such concepts as “logical consequence” as relative . . . which must, on each occasion, be related to a definite, although in greater or less degree arbitrary, division of terms into logical and extra-logical". Even Etchemendy agreed with him for once. Quine thought that analytic in a language can only be defined by fiat, and analytic as such not at all, etc. – Conifold Jun 26 at 21:09
  • @Conifold: In the "interpretation" approach, our "fiat" is limited to the definitions of logical connectives such as "and," "or," "not," and so on - everything else can then be shown using truth tables and the like. Arguably, we may define terms however we like, so this "fiat" is not actually a problem. But there remains some question of how you can "ground" an argument for the validity of logic, without using logic to do so. – Kevin Jun 27 at 2:22
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    @Conifold The idea that a logical constant can be understood as a term that is characterized entirely by the inferential relationships that it participates in is defended by Christopher Peacocke “Understanding Logical Constants: A Realist’s Account,” in Smiley & Baldwin (eds.), Studies in the Philosophy of Logic and Knowledge. OUP, p. 163 (2004) and Ian Hacking “What is Logic?” Journal of Philosophy 76, 285–319 (1979). It seems that Popper also had a similar idea in mind in “Logic Without Assumptions,” Proceedings of the Aristotelian Society 47, 251–292 (1946). – Bumble Jun 27 at 2:31
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    ... Also, as you are aware, some make appeal to logical harmony (e.g. Dummett), i.e. that logical constants are characterized by inferential rules that are normalizable and conservative with respect to what can be inferred. – Bumble Jun 27 at 2:32
  • Motivated by this interesting answer, i have added an edit in the original question. – exp8j Jun 27 at 9:15

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